Two solutions for nonhomogeneous Klein-Gordon equations coupled with Born-Infeld type equations

Abstract

This article concerns the nonhomogeneous Klein-Gordon equation coupled with a Born-Infeld type equation,$$\displaylines{- \Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u)+h(x), \quad x\in \mathbb{R}^3,\cr \Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad x\in \mathbb{R}^3, }$$ where \(\omega\)  is a positive constant. We obtain the existence of two solutions using the Mountain Pass Theorem, and the Ekeland's variational principle in critical point theory.